Even fewer than that, since you’re not accounting for the actual rules of the game. You counted every possible arrangement of X’s and O’s on the board, but many of those aren’t valid game states, like all X’s for example.
On top of that you can also eliminate rotationally equivalent states. Ditto for mirrored states. Starting with an X in the top-right isn’t a meaningfully different state than starting in any other corner. There are effectively only three distinct starting states. Center, any corner, or any side.
On the other hand, there are semi-filled final states you’re not considering. Not every square on the board needs to be filled for a player to win. You’re also only counting distinct winning lines (many of which food be eliminated due to rotational equivalence), but not the turns to get there, which would provide several possible scenarios for a given final state.
All that said, I expect the actual number of unique possible games to be quite a bit lower than 500.
There are only like 500 losing tictac toe scenarios max.
Three positions for each square (X, O, or blank), 9 squares: 3^9 = 19,683 possible game states.
Of those there are only 512 combinations where the board is compete: 2^9 = 512
Of those 512, only 16 combinations results in a win for either player. Meaning there are only 8 losing scenarios and 496 stalemate scenarios.
Even fewer than that, since you’re not accounting for the actual rules of the game. You counted every possible arrangement of X’s and O’s on the board, but many of those aren’t valid game states, like all X’s for example.
On top of that you can also eliminate rotationally equivalent states. Ditto for mirrored states. Starting with an X in the top-right isn’t a meaningfully different state than starting in any other corner. There are effectively only three distinct starting states. Center, any corner, or any side.
On the other hand, there are semi-filled final states you’re not considering. Not every square on the board needs to be filled for a player to win. You’re also only counting distinct winning lines (many of which food be eliminated due to rotational equivalence), but not the turns to get there, which would provide several possible scenarios for a given final state.
All that said, I expect the actual number of unique possible games to be quite a bit lower than 500.
Good point. There’s only 126 filled arrangements that are valid game states.
And only 3 losing states
By my estimation, of those 126: 80 are a win for X (or whoever goes first), 30 are a win for O (or whoever goes second), and 16 are a stalemate.
So the number of losing positions depends on whether you go first or second.
See, its you guys who should be up there on the cooling towers spray painting a big fat middle finger to trump and his nazis.!
Plans are in motion. See you at 50501 on Wednesday.